Distance spectral radius conditions for edge-disjoint spanning trees and a forest with constraints

Let $k\ge 2$ be a positive integer and let $G$ be a simple graph of order $n$ with minimum degree $δ$. A graph $G$ is said to have property $P(k, d)$ if it contains $k$ edge-disjoint spanning trees and an additional forest $F$ with edge number $|E(F)| > \frac{d-1}{d}(n-1)$, such that if $F$ is not a spanning tree, then $F$ has a component with at least $d$ edges. Let $D(G)$ be the distance matrix of $G$. We denote $ρ_D(G)$ as the largest eigenvalue of $D(G)$, which is called the distance spectral radius of $G$. In this paper, we investigate the relationship between the distance spectral radius and the property $P(k, δ)$. We prove that for a connected graph $G$ of order $n \ge 2k+8$ with minimum degree $δ\ge k+2$, if $ρ_D(G) \le ρ_D(K_{k-1} \vee (K_{n-k} \cup K_1))$, then $G$ possesses property $P(k, δ)$. Furthermore, for a connected balanced bipartite graph $G$ of order $n \ge 4k+8$ with minimum degree $δ\ge k+2$, we show that if $ρ_D(G) \le ρ_D(K_{\frac{n}{2}, \frac{n}{2}} \setminus E(K_{1, \frac{n}{2}-k+1}))$, then $G$ also possesses property $P(k, δ)$. Our results generalize the work of Fan et al. [Discrete Appl. Math. 376 (2025), 31--40] from the existence of $k$ edge-disjoint spanning trees to the more refined structural property $P(k, δ)$.

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